1. Field of the Invention
The present invention relates to a numerical simulation technique known as a finite-difference simulation and, more particularly, to a method of generating a discretization grid necessary for finite-difference simulation.
2. Description of the Prior Art
A numerical simulation technique known as finite-difference simulation is particularly useful in analyzing Voltage-Current characteristic and response characteristic of semi-conductor devices (See, for example, S. Selberherr "Analysis and Simulation of Semiconductor Devices" Springer-Verlag, Wien, 1984).
In the finite-difference simulation, a discretization grid is superimposed on the cross-sectional figure of a semi-conductor device to be simulated, as shown in FIG. 1. Here, in order to improve accuracy of simulation, separations between adjacent intersections of a vertical line and a horizontal line (referred hereafter simply as grid points) are reduced near boundaries of different materials such as a boundary 3 between semi-conductor base 1 and insulator medium 2, or peripheral region 5 of the electrode medium 4. The discretization grid is generated such that grid points lie on these boundaries. The physical quantities such as potentials, electron densities etc. are calculated on these grid points, and prediction or evaluation of the Voltage-Current characteristic, the response characteristic etc. are derived from these simulated quantities. The discretization grid used for finite-difference simulation most commonly is a rectangular grid such as the one shown in FIG. 1.
However, as semi-conductor devices become more highly integrated, a cross-sectional figure of the device becomes more complicated such that boundaries between different materials may include segments which are not parallel to the rectangular grid. Though it is possible to simulate such a semi-conductor device with a complex cross-sectional figure by the conventional finite-difference simulation using a rectangular grid, as shown in FIG. 2 where the grid is generated so that there is a grid point over each vertex of the boundary 6, it is extremely difficult to generate such a grid which satisfies all of the following requirements:
(a) there is a grid point over each vertex of the cross-sectional figure;
(b) there is a grid point where a line of the rectangular grid crosses a segment of the cross-sectional figure which is not parallel to the grid;
(c) separations between adjacent grid points vary gradually without a radical change, and the grid is finer around boundaries of different materials;
(d) the grid can be generated automatically.
In addition, it is very difficult to generate enough grid points on segments of the figure not parallel to the rectangular grid so as to maintain satisfactory accuracy of the simulation.
To cope with this situation, a method has been proposed in which the cross-sectional figure of the device is covered by triangles such that each vertex of the cross-sectional figure coincide with a vertex of a triangle, as shown in FIG. 3 (See, for example, S.E. Laux and R.J. Lomax "Numerical Investigation of Mesh Size Convergence Rate of the Finite Element Method in MESFFT Simulation" Solid State Electronics Vol. 24, P485, 1981). In this method, complicated figures like a boundary surface 7 and electrodes 8 can be simulated.
However, this method has the following problems:
(a) an obtuse triangle must be avoided in order to maintain the high accuracy of the simulation;
(b) it is very difficult to generate a triangular grid automatically which has grid points over all verticles of the cross-sectional figure, and which is finer around boundaries of different materials and becomes coarser gradually elsewhere;
(c) calculations required for the simulation with a triangular grid involve matrices in which non-zero elements appears irregularly. For example, a configuration shown in FIG. 4(A) requires calculations involving a matrix shown in FIG. 4(B) where a non-zero element is represented by star symbols. This requires a computer with a larger memory capacity than that normally required for a simulation with a rectangular grid, and consequently renders the method impractical.